If I am not mistaken, the SOLVE and INTEGRATE functions had Professor Kahan as it's mentor. Kahan's philosophy was to push the accuracy to the machine limits and not by a user-defined value.

The HP-15C Advanced Functions Handbook has a section "Accuracy of the Function to be Integrated" that specifies the accuracy of the integral, which is basically the accuracy of the function you are integrating. This tells me that if you are integrating a function using an empirical approximation, then expect more error in the integral.